Andrew Fanoe | Searcharxiv

arXiv2012

Hamiltonian $S^1$ actions with Isolated Fixed Points on 6-Dimensional Symplectic Manifolds

The question of what conditions guarantee that a symplectic $S^1$ action is Hamiltonian has been studied for many years. In a 1998 paper, Sue Tolman and Jonathon Weitsman proved that if the action is semifree and has a non-empty set of isolated fixed points then the action is Hamiltonian. Furthermore, in a 2010 paper Cho, Hwang, and Suh proved in the 6-dimensional case that if we have $b_2^+=1$ at a reduced space at a regular level $λ$ of the circle valued moment map, then the action is Hamiltonian. In this paper, we will use this to prove that certain 6-dimensional symplectic actions which are not semifree and have a non-empty set of isolated fixed points are Hamiltonian. In this case, the reduced spaces are 4-dimensional symplectic orbifolds, and we will resolve the orbifold singularities and use J-holomorphic curve techniques on the resolutions.

math.SG↗

arXiv2012

Toric Structures on Symplectic Bundles of Projective Spaces

Recently, extending work by Karshon, Kessler and Pinsonnault, Borisov and McDuff showed that a given symplectic manifold $(M,ω)$ has a finite number of distinct toric structures. Moreover, McDuff also showed a product of two projective spaces $\bC P^r\times \bC P^s$ with any given symplectic form has a unique toric structure provided that $r,s\geq 2$. In contrast, the product $\bC P^r \times \bC P^1$ can be given infinitely many distinct toric structures, though only a finite number of these are compatible with each given symplectic form $ω$. In this paper we extend these results by considering the possible toric structures on a toric symplectic manifold $(M,ω)$ with $\dim H^2(M)=2$. In particular, all such manifolds are $\bC P^r$ bundles over $\bC P^s$ for some $r,s$. We show that there is a unique toric structure if $r<s$, and also that if $r,s\geq 2$ then $M$ has at most finitely many distinct toric structures that are compatible with any symplectic structure on $M$. Thus, in this case the finiteness result does not depend on fixing the symplectic structure. We will also give other examples where $(M,ω)$ has a unique toric structure, such as the case where $(M,ω)$ is monotone.

math.SG↗